Vector Space

Vector Space Definition

A vector space consists of a set of V ( elements of V are called vectors), a field F ( elements of F are scalars) and the two operations

  • Vector addition is an operation that takes two vectors u, v ∈ V, and it produces the third vector u + v ∈ V
  • Scalar Multiplication is an operation that takes a scalar c ∈ F and a vector v ∈ V and it produces a new vector uv ∈ V.

Where both the operations must satisfy the following condition

Elements of V are mostly called vectors and the elements of F are mostly scalars. There are different types of vectors. To qualify the vector space V, the addition and multiplication operation must stick to the number of requirements called axioms. The axioms generalise the properties of vectors introduced in the field F. If it is over the real numbers R is called a real vector space and over the complex numbers, C is called the complex vector space.

Vector Space Axioms

All the axioms should be universally quantified. For vector addition and scalar multiplication, it should obey some of the axioms. Here eight axiom rules are given.

Conditions for Vector Addition

An operation vector addition ‘ + ‘ must satisfy the following conditions:

Closure : If x and y are any vectors in the vector space V, then x + y belongs to V

  • Commutative Law : For all vectors x and y in V, then x + y = y + x
  • Associative Law : For all vectors x, yand z in V, then x + ( y + z ) = ( x + y ) + z
  • Additive Identity : For any vector x in V, the vector space contains the additive identity element and it is denoted by ‘ 0 ‘ such that 0 + x = x and x + 0 = x
  • Additive inverse : For each vector x in V, there is an additive inverse -x to get a solution in V.

Condition for Scalar Multiplication

An operation scalar multiplication is defined between a scalar and a vector and it should satisfy the following condition :

Closure: If x is any vector and c is any real number in the vector space V, then x. c belongs to V

  • Associative Law: For all real numbers c and d, and the vector x in V, then c. (d. v ) = ( c . d). v
  • Distributive law: For all real numbers c and d, and the vector x in V, (c + d). v = c . v + c . d
  • Distributive law: For all real numbers c and the vectors x and y in V, c. ( x + y) = c. x + c. y
  • Unitary Law : For all vectors x in V, then 1. v = v . 1 = v

Vector Space Properties

Here are some basic properties that are derived from the axioms are

    • The addition operation of a finite list of vectors v1 v2, . . . , vk can be calculated in any order, then the solution of the addition process will be the same.
    • If x + y = 0, then the value should be y = −x.
    • The negation of 0 is 0. This means that the value of −0 = 0.
    • The negation or the negative value of the negation of a vector is the vector itself: −(−v) = v.
    • If x + y = x, if and only if y = 0. Therefore, 0 is the only vector that behaves like 0.
    • The product of any vector with zero times gives the zero vector. 0 x y = 0 for every vector in y.
    • For every real number c, any scalar times of the zero vector is the zero vector. c0 = 0
    • If the value cx= 0, then either c = 0 or x = 0. The product of a scalar and a vector is equal to when either scalar is 0 or a vector is 0.
    • The scalar value −1 times a vector is the negation of the vector: (−1)x = −x. We define subtraction in terms of addition by defining x − y as an abbreviation for x + (−y).

x − y = x + (−y)

All the normal properties of subtraction follows

  • x + y = z then the value x = z − y.
  • c(x − y) = cx − cy.
  • (c − d)x = cx − dx

Vector Space Problems

Go through the vector space problem provided here.

Question :

Show that each of the conditions provided is in vector space

  1. The set of linear polynomials \(p_{1}=\left \{ a_{0}+a_{1}x|a_{0},a_{1}\epsilon \mathbb{R} \right \}\) under the addition and scalar multiplication operations
  2. Under usual matrix operation, the set of 2 x 2 matrix with real entries
  3. Three component row vectors with usual operations
  4. The set A = \(\left \{ \begin{pmatrix} x\\ y\\ z\\ w\end{pmatrix}\epsilon \mathbb{R}^{4}|x+y-z+w=0 \right \}\) under the operations from\(\mathbb{R}^{4}\)

Solution :

Conditions checked from the axioms and properties:

  1. The zero element is 0 + 0x is zero
  2. The zero element of the vector space under 2 x 2 matrix is zero
  3. The zero element of three component row vectors are zeros
  4. The closure property of addition involves \(\begin{pmatrix} x_{1}\\ y_{1}\\ z_{1}\\ w_{1} \end{pmatrix}+\begin{pmatrix} x_{2}\\ y_{2}\\ z_{2}\\ w_{2} \end{pmatrix}=\begin{pmatrix} x_{1}+x_{2}\\ y_{1}+y_{2}\\ z_{1}+z_{2}\\ w_{1}+w_{2} \end{pmatrix}\) is in A because (x1+x2) + (y1+y2) – (z1+z2) + (w1+w2) = ( x1+y1-z1+w1)+(x2+y2-z2+w2) = 0 + 0

Similarly, the closure of scalar multiplication can be obtained.

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